On some modules of covariants for a reflection group

Abstract:

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak {h}$ and Weyl group $W$. We build up a graded isomorphism $\bigl (\bigwedge \mathfrak {h}\otimes \mathcal H\otimes \mathfrak {h}\big )\vphantom )^W\to \bigl (\bigwedge \mathfrak {g}\otimes \mathfrak {g}\big )^{\mathfrak {g}}$ of $\bigl (\bigwedge \mathfrak {g}\big )^{\mathfrak {g}}\cong S(\mathfrak {h})^W$-modules, where $\mathcal H$ is the space of $W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation.